Optical-coupled resonator structures based on loop-coupled cavities and loop coupling phase

ABSTRACT

A resonator structure includes an input waveguide and an output waveguide. In one embodiment, the resonator structure also includes at least one resonator that couples the input waveguide to the output waveguide and a directional coupler that optically couples the input waveguide to the output waveguide. In another embodiment, the resonator structure includes a plurality of ring resonators that couple the input waveguide to the output waveguide. The plurality of ring resonators include a sequence of ring resonators that form a coupling loop. Each ring resonator in the sequence is coupled to at least two other ring resonators in the sequence and the first ring resonator in the sequence is coupled to the last ring resonator in the sequence so as to form the coupling loop.

TECHNICAL FIELD

The invention generally relates to optical resonator structures. Moreparticularly, the invention relates to integrated optical(microphotonic) resonant structures that comprise loop-coupled cavitieshaving an associated loop coupling phase, and may include, for example,microring resonators, photonic crystal microcavities, and standing wavedielectric resonators.

BACKGROUND

Coupled resonators have previously been used in optics, and particularlyin integrated optics, for the design of various amplitude and phasefiltering structures. Previous work on coupled-cavity resonantstructures includes: series-coupled-cavity (SCC) structures (FIG. 1),parallel-coupled-cavity (PCC) filters (FIG. 2), and Mach-Zehnderall-pass decomposition (AD) resonant structures (FIG. 3).

Series-coupled-cavity structures (also referred to as coupled-resonatoroptical waveguides—CROWs—in slow-light literature), such as thestructure 310 shown in FIG. 1, may be used as channel add-drop filtersand slow-wave structures. The structure 310 includes a chain of linearlycoupled optical cavities 311-313, an input waveguide 314 coupled to thefirst cavity 31, and an output waveguide 315 coupled to the last cavity313. A drawback of SCC structures is that they only support all-polespectral responses in the input-to-drop-port response, with no controlover the transmission zeros (all located at effectively infinitefrequency detuning from the center wavelength). When making a comparisonfor a given resonant order (i.e., a fixed number of used cavities, orused resonant modes, in the structure), all-pole responses, such asmaximally-flat (Butterworth) and equiripple (Chebyshev), are known incircuit theory to be suboptimal in their selectivity in comparison topole-zero filters, such as elliptic and quasi-elliptic designs. Anotherdrawback is that such all-pole structures are minimum-phase and are, asis well known in signal processing, constrained to have their amplitudeand phase response uniquely related by the Kramers-Kronig (Hilberttransform) relation. This means that a flat-top amplitude responseimplies a dispersive phase response, and high-order, selective(square-passband) filters are as a result highly dispersive. Thedispersion can substantially distort an optical signal passing throughthe filter, so additional dispersion compensating (all-pass filter)structures must ordinarily follow such filters to permit reasonably highspectral efficiency.

With reference to FIG. 2, parallel-coupled-cavity structures 320 havebeen used as channel filters and for demonstration of light slowing. Thestructure 320 includes a pair of parallel optical waveguides 321, 322and a set of optical resonant cavities 323-325. Each of the cavities323-325 is typically individually coupled to the input waveguide 321 andto the output waveguide 322, while no one of the cavities 323-325 issubstantially coupled directly to another cavity 323-325.Parallel-coupled-cavity structures 320 also permit flat-top filterresponses in the input-to-drop-port response, i.e., where an inputsignal enters one waveguide and an output (dropped-wavelength) signalexits from the second waveguide. A drawback of these structures 320 isthat there is only one resonator 323-325 at any point separating theinput waveguide 321 and output waveguide 322 (in the sense of opticalcoupling), and higher-order responses may be obtained due tophase-aligned constructive interference of the many cavities 323-325. Ifsubstantial variations in the design parameters are introduced due tofabrication errors, the drop port response rejection can be severelyreduced, and in the limit of substantial variations may approach afirst-order rolloff. A second drawback is that parallel-coupled-cavitystructures 320 have interferometric paths between the cavities 323-325and, as such, their resonant wavelengths are difficult to tune whilemaintaining the phase relationships of the interferometric paths. Theparallel-coupled-cavity structures 320 are also difficult to switch onand off in a hitless manner.

With reference to FIG. 3, a typical Mach-Zehnder all-pass decomposition(AD) structure 330 includes a Mach-Zehnder (MZ) interferometer formed oftwo waveguides 331, 332 having a first 3 dB coupler 333 and a second 3dB coupler 334, and resonant cavities 335 in an all-pass configurationin one or the other waveguide 331, 332 inside the interferometer, i.e.,between the first and the second 3 dB couplers 333, 334. One advantageof the structure 330 over an SCC filter, such as the filter 310illustrated in FIG. 1, is that the structure 330 permits optimallysharp, “elliptic” filter response designs. A drawback of these designs,however, is that they require a precise 3 dB coupling (50:50% splitting)in each directional coupler 333, 334, at all wavelengths in thewavelength range of operation. Deviation from 3 dB coupling leads toreduction of the signal rejection ratio in the drop port 336, whichappears as a flat “noise floor” because a fraction of light at allwavelengths ventures into the drop port 336. This requirement makes suchfilters challenging to fabricate and control in practice, as theyrequire an accurate 3 dB coupling over the entire spectrum of operationof the filter (in and out of band).

Referring to FIG. 4, a SCC structure 40 based on standing-waveresonators 41-44 is shown, as known in the prior art. The input port 45and through port 46 are the incident (incoming) and reflected (outgoing)waves in the top waveguide, respectively, while the drop port 47 is theoutgoing wave in the bottom waveguide. An important advantage ofmicroring resonator structures such as the structure 310 depicted inFIG. 1 is that they, by their traveling-wave nature, inherently separatethe incoming and outgoing waves into separate waveguide ports. Thismakes each port automatically “matched” (i.e., matched impedance), eachport having no substantial reflection signal when an incident signal issent into the port. Such structures that inherently have matched ports(as in FIG. 1) can be called “optical hybrids”, and they eliminate theneed for optical isolators and circulators.

SUMMARY OF THE INVENTION

The optical structures of the present invention solve the problem ofproviding transmission zeros, with positions in frequency (or, moregenerally, on the complex-frequency plane) controllable in design, inthe drop-port response(s) of a resonant structure. The inventive opticalstructures do not require the use of 3 dB couplers or any strong directcoupling between the input and output waveguide, and therefore havespectral responses, including rejection ratios, that are highlyinsensitive to fabrication errors and design nonidealities, in contrastto AD structures. The inventive structures also allow non-minimum-phaseresponses, including flat-top, linear-phase filters, unlike SCCstructures which are limited to all-pole responses.

In one embodiment, the structures of the present invention, whichinclude at least an input port and a drop port, enable the design ofspectral responses with transmission zeros in the drop port (at real andcomplex frequency detunings from resonance). This enables optimallysharp filter responses (using real-frequency zeros), anddispersion-engineered spectral responses (using complex-frequencyzeros). The latter responses are non-minimum-phase and therefore are notsubject to the well known Kramers-Kronig (Hilbert transform) constraintbetween the amplitude and phase spectral responses. In particular, thestructures of the present invention permit the design of spectralresponses with passbands having a flat-top amplitude response at thesame time as a nearly linear phase response, without the need foradditional dispersion compensation following the structure (e.g., byall-pass filters). Such responses are optimal, in the sense that aminimum number of resonant cavities are used for a given amplituderesponse selectivity and phase linearity.

Applications of the disclosed structures include: i) channel add-dropfilters for high spectral efficiency photonic networks fortelecommunication applications as well as for intrachip photonicnetworks for next-generation microprocessors; ii) dispersion-compensatedfilters; iii) slow-wave resonator-based structures for sensors,channelized modulators, amplifiers, wavelength converters, andcoupled-cavity nonlinear optics in general. Another example applicationis in microwave photonics, where the flat-top, linear-phasemicrophotonic filters may be used in combination with an opticalmodulator to replace microwave satellite transponder filters, therebyreducing the size and weight of the payload. The flat-top, linear-phasemicrophotonic filters may also be used in terrestrial microwavefiltering, such as in spectral slicing filters in cellular telephonetowers.

The optical structures of the present invention generally include one ormore optical cavities and at least two waveguides. In one embodiment, aninput port and a through (reflection) port are defined in the firstwaveguide, and a drop (transmission) port is defined in the secondwaveguide. Each of the first and the second waveguides may be coupled toat least one optical cavity, and the direct coupling between the firstand second waveguides, i.e., the optical power coupled at a wavelengthfar from the resonance frequency of any optical cavity in the system,is, in one embodiment, less than about 50%. Preferably, the directcoupling between the first and second waveguides is less than about 10%.However, the direct coupling between the first and second waveguides mayor may not be substantially negligible, as dictated by the particulardesign.

In various embodiments, the optical structures of the present inventioninclude one or more of the following features:

-   -   1) The use of cavity modes having at least one node in the        spatial electric field pattern at any one time (referred to        herein as “high-order cavity modes”).    -   2) A plurality of cavities coupled in a loop, thereby forming a        “coupling loop” and defining an associated loop coupling        coefficient (LCC).    -   3) For structures exhibiting feature 2), a further defined phase        of the LCC, referred to as the loop coupling phase (LCP). The        LCP of the inventive structures may be approximately 0,        approximately 180 degrees, or any other value between 0 and 360        degrees. The LCP may be chosen by an appropriate arrangement of        the cavity mode geometry. In the case of microring resonators,        the LCP may be chosen by tilting the geometry of the coupled        cavity loop by an appropriate angle in the plane of the        resonators.    -   4) A non-zero (non-negligible) direct coupling between the input        waveguide and the output waveguide, but still one that is less        than about 50%, and is preferably less than about 10%, and even        more preferably is less than about 1%.

The development of optical filter designs based on a family of resonatorstructures incorporating one or more of the above features, and usingappropriate energy coupling coefficients between resonant cavities andbetween cavities and ports, permits the realization of various filterresponses having transmission zeros. For example, optimally sharpfilters, including elliptic and quasi-elliptic filters, and dispersionengineered filters, including nearly linear phase filters, may berealized.

In general, in one aspect, the invention features a loop-coupledresonator structure that includes an input waveguide, an outputwaveguide, and a plurality of ring resonators that couple the inputwaveguide to the output waveguide. The plurality of ring resonatorsinclude a sequence of ring resonators that form a coupling loop. Eachring resonator in the sequence is coupled to at least two other ringresonators in the sequence and the first ring resonator in the sequenceis coupled to the last ring resonator in the sequence so as to form thecoupling loop. As used herein, the term “ring resonator” generallyrefers to any resonator that is formed by wrapping a waveguide into aclosed loop. Accordingly, ring resonators include, for example, circularmicroring resonators and racetrack resonators.

In various embodiments of this aspect of the invention, the couplingloop has an associated loop coupling coefficient, which itself has anassociated loop coupling phase. The loop coupling phase may beapproximately 0 degrees, approximately 180 degrees, or another amount.The coupling loop may include four ring resonators and each of the ringresonators of the coupling loop may include a substantially equal ringradii. In one embodiment, the four ring resonators forming the couplingloop are each centered at a different vertex of a rectangle. In anotherembodiment, the geometry of the coupling loop is tilted so that the fourring resonators forming the coupling loop are each centered at adifferent vertex of a parallelogram. For example, the geometry of thecoupling loop may be tilted by an angle equal to approximately ⅛ of theguided wavelength of a ring resonator in the coupling loop.

In yet another embodiment, half of the plurality of ring resonators arearranged in a first row and half of the plurality of ring resonators arearranged in a second row adjacent to the first row. In such anembodiment, each ring resonator in the first row may be coupled to atleast one other ring resonator in the first row and to a ring resonatorin the second row, and each ring resonator in the second row may becoupled to at least one other ring resonator in the second row and to aring resonator in the first row. The inter-row couplings (for example,the inter-row energy coupling coefficients) of the resonators may beweaker than the intra-row couplings of the resonators.

At least one of the plurality of ring resonators may be a microringresonator or, alternatively, a racetrack resonator. Moreover, at leastone of the plurality of ring resonators may include magnetooptic media.In various embodiments, the coupling loop includes an even number ofring resonators. In such a case, at least one ring resonator (and, morespecifically, every ring resonator) in the coupling loop is operated toonly propagate light in a single direction within the resonator. Instill another embodiment, the output waveguide includes a drop port andthe loop-coupled resonator structure has a spectral response thatincludes transmission zeros in the drop port.

In general, in another aspect, the invention features an opticalresonator structure that includes an input waveguide, an outputwaveguide, at least one resonator that couples the input waveguide tothe output waveguide, and a directional coupler that optically couplesthe input waveguide to the output waveguide.

In various embodiments of this aspect of the invention there is a phaseshift in the input waveguide light propagation between the directionalcoupler and the at least one resonator, relative to that between thedirectional coupler and the at least one resonator in the outputwaveguide. Alternatively, the relative phase shift may be introduced inthe output waveguide between the directional coupler and the at leastone resonator. Additionally, a length of the input waveguide between thedirectional coupler and a point at which the at least one resonatorcouples to the input waveguide may be substantially equal to a length ofthe output waveguide between the directional coupler and a point atwhich the at least one resonator couples to the output waveguide. Therelative phase shift introduced may be implemented, for example, as asmall waveguide length difference (e.g. half of the guided wavelengthlong, for a 180° relative phase shift) between the input and outputwaveguide, between the directional coupler and resonators, or as athermally-actuated phase shifter.

In one embodiment, a plurality of resonators couple the input waveguideto the output waveguide. The plurality of resonators may include, forexample, a sequence of resonators that form a coupling loop. Eachresonator in the sequence may be coupled to at least two otherresonators in the sequence and the first resonator in the sequence maybe coupled to the last resonator in the sequence so as to form thecoupling loop.

The optical coupling between the input waveguide and the outputwaveguide introduced by the directional coupler may be substantiallybroadband over several passband widths. For example, it may be broadbandover at least three passband widths, or over at least ten passbandwidths. In another embodiment, the output waveguide includes a drop portand the optical resonator structure has a spectral response thatincludes transmission zeros in the drop port.

These and other objects, along with advantages and features of theinvention, will become more apparent through reference to the followingdescription, the accompanying drawings, and the claims. Furthermore, itis to be understood that the features of the various embodimentsdescribed herein are not mutually exclusive and can exist in variouscombinations and permutations.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, like reference characters generally refer to the sameparts throughout the different views. Also, the drawings are notnecessarily to scale, emphasis instead generally being placed uponillustrating the principles of the invention. In the followingdescription, various embodiments and implementations are described withreference to the following drawings, in which:

FIG. 1 illustrates one embodiment of a series-coupled-cavity structureknown in the art;

FIG. 2 illustrates one embodiment of a parallel-coupled-cavity filterknown in the art;

FIG. 3 illustrates one embodiment of a Mach-Zehnder all-passdecomposition resonant structure known in the art;

FIG. 4 illustrates one embodiment of a series-coupled-cavity structure,known in the art, that is based on standing-wave resonators;

FIG. 5 a illustrates a third-order loop-coupled resonant structure withnegative LCC in accordance with an embodiment of the invention;

FIG. 5 b illustrates a third-order loop-coupled resonant structure withpositive LCC in accordance with an embodiment of the invention;

FIG. 6 a illustrates a set of four traveling-wave circular diskresonators, with positive LCC, in accordance with an embodiment of theinvention;

FIG. 6 b illustrates a set of four traveling-wave circular diskresonators, with negative LCC, in accordance with an embodiment of theinvention;

FIG. 7 illustrates a loop-coupled traveling-wave structure having anegative LCC in accordance with an embodiment of the invention;

FIG. 8 illustrates a loop-coupled traveling-wave structure having apositive LCC in accordance with an embodiment of the invention;

FIG. 9 illustrates a MZ model of a loop coupling in accordance with anembodiment of the invention;

FIG. 10 a is a graph that illustrates a frequency-normalized performanceof a particular design (choice of energy coupling coefficients) of thedevice of FIG. 7 in accordance with an embodiment of the invention;

FIG. 10 b illustrates the location in the complex-frequency plane of thetransmission zeros for several exemplary LCPs;

FIG. 11 illustrates a loop-coupled traveling-wave structure inaccordance with an embodiment of the invention;

FIG. 12 is a graph that compares three exemplary 4^(th)-order filterdesigns of different geometries;

FIG. 13 a illustrates a 14^(th)-order structure in microring-resonatortechnology in accordance with an embodiment of the invention;

FIG. 13 b illustrates a functionally equivalent structure to thestructure of FIG. 13 a, using standing-wave optical cavities, inaccordance with an embodiment of the invention;

FIG. 13 c illustrates a 30^(th)-order structure in microring-resonatortechnology in accordance with an embodiment of the invention;

FIG. 14 a illustrates one embodiment of an electric circuit that istopologically compatible with the structures depicted in FIGS. 13 a-c;

FIG. 14 b illustrates that a pair of capacitors (each representing aresonator) connected by an immitance (impedance or admittance) invertermay be equivalent to an L-C resonator;

FIG. 15 a is a graph that compares the resulting spectra of thestructure depicted in FIG. 13 c to that of a typical maximally-flat SCCfilter;

FIG. 15 b is a graph that compares the group delay response spectra ofthe structure depicted in FIG. 13 c to that of a typical maximally-flatSCC filter;

FIG. 15 c is a graph that compares the dispersion response spectra ofthe structure depicted in FIG. 13 c to that of a typical maximally-flatSCC filter;

FIG. 16 illustrates a loop-coupled resonant structure usingstanding-wave cavities in accordance with an embodiment of theinvention;

FIG. 17 a illustrates a loop-coupled resonant structure using microringresonators in accordance with an embodiment of the invention;

FIG. 17 b illustrates a symmetric waveguide loop mirror in accordancewith an embodiment of the invention;

FIG. 17 c illustrates the symmetric waveguide loop mirror of FIG. 17 bcoupled to a cavity in accordance with an embodiment of the invention;

FIGS. 18 a-c illustrate an exemplary physical realization of the4^(th)-order loop-coupled structure depicted in FIG. 7;

FIGS. 19 a-c illustrate an exemplary physical realization of a4^(th)-order standard SCC filter;

FIG. 20 is a graph that compares the simulated spectra of the devicesdepicted in FIGS. 18 a-c to that of the devices depicted in FIGS. 19a-c, without excess waveguide loss;

FIG. 21 is a graph that compares the simulated spectra of the devicesdepicted in FIGS. 18 a-c to that of the devices depicted in FIGS. 19a-c, with the effect of waveguide loss added;

FIG. 22 is a graph that illustrates the drop and through port responsesof an embodiment of the loop-coupled structure depicted in FIG. 7; thegraph also illustrates a response of a standard Butterworth(maximally-flat) all-pole filter, implementable in a SCC structure;

FIG. 23 is a graph that illustrates the response of an embodiment of a4^(th)-order Chebyshev filter, using a SCC structure;

FIG. 24 is a graph that compares the group delay spectra of anembodiment of a 4^(th)-order Chebyshev filter, using a SCC structure, tothat of the loop-coupled structure depicted in FIG. 7;

FIG. 25 is a graph that compares the dispersion spectra of an embodimentof a 4^(th)-order Chebyshev filter, using a SCC structure, to that ofthe loop-coupled structure depicted in FIG. 7;

FIG. 26 illustrates a second-order filter in accordance with anembodiment of the invention;

FIG. 27 is a graph that compares the drop and through responses of astandard SCC filter to those of the second-order filter depicted in FIG.26;

FIGS. 28-31 illustrate loop-coupled resonator based optical filtersusing standing wave resonators in accordance with various embodiments ofthe invention; and

FIGS. 32-37 illustrate loop-coupled resonator based optical filtersusing standing wave cavities in accordance with various embodiments ofthe invention.

DETAILED DESCRIPTION

In general, the present invention pertains to optical-coupled resonatorstructures that are based on loop-coupled cavities and loop couplingphase.

Loop coupling of cavities may be understood as follows. FIGS. 5 a-b eachshow one embodiment of a third-order loop-coupled resonant structure 50(i.e., a resonant structure having three cavities or, more rigorously,three resonant modes in use near the wavelength of interest, which mayor may not be substantially located in separate cavities), based onstanding wave cavities. The structure 50 includes a single couplingloop. A (non-trivial) coupling loop is defined as a path through asequence of mutually coupled cavities that begins with an arbitrarycavity and that ends at that same arbitrary cavity after traversing atleast two other cavities, but without the path being retraced. Forexample, the coupling loop of FIGS. 5 a and 5 b can be written as thesequence 52-54-56-52, with numbers referring to the cavities. Sequencesrepresenting non-trivial coupling loops are not palindromes, i.e., arenot symmetric with respect to the middle number.

For each coupling loop, a loop coupling coefficient (LCC) may bedefined. Two cavity modes may be coupled via evanescent field couplingacross a gap or via radiative field coupling via a traveling-wavecoupling pathway. Evanescent coupling may be understood as follows. Foreach pair of mutually coupled cavities, an energy coupling coefficient(ECC), with units of rad/s, represents the rate of energy amplitudecoupling in time between the two cavities, and is the standard measureof coupling between cavities under the commonly used coupled mode theoryin time formalism. Then, the LCC of a coupling loop is defined as theproduct of ECCs around the loop. Strictly, the LCC is a vector, with acomplex magnitude indicating the product of ECCs, and whose directionindicates the sense in which the loop is traversed according to a presetconvention, for example according to the well-known right-hand-rule usedin electrical engineering. In the following discussion, the direction ofthe loop is understood from the illustrations and is omitted in the LCCvalue, where only the magnitude is considered. For example, if the ECCfrom cavity i to cavity j is μ_(i,j), then the LCC for the structure 50depicted in FIGS. 5 a and 5 b is μ_(54,52) μ_(56,54) μ_(52,56).

Since the LCC is a complex number, a loop-coupling phase (LCP)associated with each LCC may be defined as the phase of the LCC. As withany phase, the LCP has a value modulo 360°, i.e., modulo 2π radians. TheLCP has a physical significance, which is illustrated in FIGS. 5 a and 5b. FIG. 5 a shows a coupling loop with a 180° (π radian) LCP, which isalso referred to as negative loop coupling. The LCP can be ascertainedby assuming the illustrated two-lobe electric field patterns for thecavity modes 52, 54, 56 (showing areas of positive and negative electricfield), with a choice of reference time that makes the field phasordistribution real. Overlap integrals of modes of adjacent cavities withthe (generally positive) dielectric distribution perturbation gives theECCs, μ_(i,j), all to be negative real numbers, and thus their product,the LCC, is also negative (because there is an odd number of cavities52, 54, 56, and thus an odd number of ECCs, μ_(i,j)).

It should be noted that in the microwave engineering literature,reference is often made to a positive or negative couplingcoefficient—this refers to the sign of each ECC, μ_(i,j). Since the signof each cavity's (uncoupled) resonant mode pattern is arbitrary, one ofthe cavity field distributions may be multiplied by −1 in order tochange the sign of two ECCs—the two associated with the two couplings tothat cavity. Hence, the sign of each ECC is not unique. However, the LCPis not changed and is an invariant property of the structure withrespect to the arbitrary choice of spatiotemporal reference phase foreach cavity mode. When ECCs μ_(i,j) are substantiallywavelength-independent near resonance, as in the case of evanescentcoupling, then the LCP is also not a function of wavelength. Therefore,positive and negative coupling coefficients may be referred to, but itshould be understood that these are not unique definitions, and the LCP(set) defines the structure properties uniquely, to within an additionalport-to-port transmission phase. It turns out that positive, negative,and arbitrary-phase LCCs enable different response functions to beachieved, as described further below. Therefore, it is important todesign physical structures that are capable of realizing “positive,”“negative,” and arbitrary-phase coupling coefficients, such that theirproduct, the LCC, may also be such, i.e., positive, negative, orarbitrary-phase complex.

The Applicant has observed that it is possible to achieve negativecoupling in optical resonators by using high-order resonant modes, i.e.,modes with at least one null in the electric field pattern. Then, thepositive and negative lobe of the same resonant mode may be coupled tovarious other modes, and a negative LCC may also be generated, as donein FIG. 5 a. While in microwave engineering typically low-order resonantmodes are used due to size, weight, and loss Q constraints with metalliccavities, in optical resonators using high-order cavity modes is not adisadvantage since optical resonators often operate at high-order toachieve high Q (e.g., microring resonators). This is because dielectricoptical resonators are confined by refractive index discontinuity, andradiation Q can be increased by increasing cavity size and mode order.

High-order (i.e., second-order) field pattern resonances are used inFIG. 5 a to produce a negative LCC. In FIG. 5 b, it is shown that ageometrical rearrangement of the resonant mode coupling configurationpermits positive LCC in the same structure, by forming two negativecouplings (μ_(54,52) and μ_(52,56)) and one positive coupling(μ_(56,54)). The strengths (magnitudes) of all coupling coefficients mayremain the same in the two configurations. When external access portsare coupled to the resonant system, the phase of the LCC (i.e., the LCP)has unique consequences for the response. With an appropriate choice ofcoupling coefficients, a third-order bandpass filter may be formed witha single transmission zero in the drop port response. Then, the cases ofpositive and negative LCC have the transmission zero on opposite sidesof the passband when all coupling magnitudes are kept fixed, but the LCPis changed by 180°.

Electromagnetic reciprocity restricts the LCPs that may be obtained in aresonant system. Structures formed of reciprocal media, if alsolossless, have a uniform phase across the cavity field pattern, whichcan thus be (for standing-wave-cavity modes) assumed a real numberdistribution with a particular choice of reference phase. Hence, onlypositive or negative coupling coefficients, and thus only a positive ornegative LCC, or 0 or 180° LCP, may be produced. An exception isdegenerate resonators, which are addressed below.

Non-reciprocal resonators, which include magneto-optic media, as well asreciprocal traveling-wave resonators, permit arbitrary-phase couplingand thus permit an arbitrary LCP. Non-reciprocal lossless structures donot have the restriction of uniform phase across the resonant modefield. Reciprocal traveling-wave (including microring) resonators behavethe same way because they have two degenerate resonances. Either twodegenerate, orthogonal standing-wave modes (excited 90° out of phase tosimulate a traveling-wave resonance), or two (a clockwise andanti-clockwise) propagating traveling-wave modes may be chosen as abasis. When the latter point of view is chosen, arbitrary LCPs may beobtained. Whether LCPs can be arbitrary depends on the choice of basisbecause the LCC depends on the ECCs, which in turn are defined withrespect to the resonant modes; so, the choice of basis of resonant modesis ultimately that with respect to which the LCP has a meaning. This isexplained in greater detail below.

FIGS. 6 a and 6 b each show a set 60 of four traveling-wave circulardisk resonators 62, 64, 66, 68 (which may, alternatively, be ringresonators), each having an excited resonance that has two wavelengthsround-trip. FIG. 6 a depicts a positive LCC, while FIG. 6 b depicts anegative LCC. In order to maintain unidirectional resonance excitation(i.e., excite 1 mode per cavity), coupling loops using an even number ofcavities are, in one embodiment, formed, because the sense (angulardirection) of propagation changes from cavity to cavity. One way to seehow the loop coupling coefficient is formed is to look at one of the twosets of standing wave modes that are excited 90° out-of-phase tosimulate the traveling-wave resonance. One set of standing-wave modes(modes of the individual, uncoupled cavities considered), chosen so asto be orthogonal and not couple to the second set, is shown in FIG. 6 a.By observing the overlap integrals, all four positive by inspection, itmay be ascertained that FIG. 6 a, with a square-grid arrangement of ringor disk resonators 62, 64, 66, 68, provides a positive LCC, i.e., 0°LCP. FIG. 6 b shows a snapshot of the same system of four cavities 62,64, 66, 68, with modes oriented to obtain three positive couplingcoefficients (μ_(64,62), μ_(66,64), and μ_(68,66)) and one negativecoupling coefficient (μ_(62,68)), resulting in a negative LCC, i.e.,180° LCP. To obtain a negative LCC, the square-grid ring resonatorarrangement may be tilted by ⅛^(th) of the guided wavelength in theresonator. In this case, since each resonator 62, 64, 66, 68 has 2wavelengths per round-trip, one wavelength covers a 180° angle of motionaround the cavity and the tilt is thus 180°/8=22.5°. This assumes thatall four coupling coefficients are equal, i.e., that the coupling gapsare all the same. In general, the coupling coefficients and couplinggaps may all be different, and the geometrical tilt may be differentfrom ⅛^(th) wavelength.

A second way to obtain the value of the LCP is to begin at any resonator62, 64, 66, 68 and follow the propagation path of a traveling waveexcited in the resonator 62, 64, 66, 68, traversing all four resonators62, 64, 66, 68 (in any way that makes a full traversal), and returningto the starting point after a full round trip. In such a case, the LCPis the total propagation phase accumulated in the traversal, assumingfor each cavity its propagation constant at its uncoupled resonancefrequency, and not applying any additional phase at points wherecrossing from one cavity to the next via a directional coupler.

FIGS. 6 a and 6 b also set forth a physically helpful naming of thepositive and negative couplings. If mode fields are unipolar (no nodes)and taken by convention as positive, a positive coupling results from apositive (electric) polarizability of the coupling perturbation. Sincethe positive coupling perturbation stores electric energy, it may alsobe referred to as capacitive coupling. Negative coupling, likewise,results from a negative (magnetic) polarizability of the perturbation,and may also be referred to as inductive coupling. The ⅛ wavelength tiltin the traveling wave resonators in a 4-cavity loop, or the use of astanding-wave mode with a field null, to get negative coupling suggestsa physical interpretation of the geometrical as an impedancetransformation applied to one positive coupling coefficient. A negative(inductive) coupling may be thought of as a positive (capacitive)coupling with a quarter-wave impedance transformer on each end, which isequivalent to a sign flip (180° phase change) in the field of one of themodes.

Finally, since tilting the geometry of coupled microring (or othertraveling-wave) cavities may change the LCC from positive to negative,tilts other than ⅛^(th) of a guided wavelength may result inarbitrary-phase LCC and thus arbitrary LCP. This is possible in rings,while it typically is not possible in reciprocal standing-wave cavities.For arbitrary LCPs in general, one set of orthogonal standing wave modesin the rings depicted in FIG. 6 a and the second, complementary set oforthogonal modes are not independent, but are rather coupled to eachother when placed in a loop. Therefore, arbitrary LCP may be simulatedwith two sets of standing-wave cavities appropriately coupled.

Now, several exemplary loop-coupled structures with various LCPconfigurations are shown and their properties described.

FIG. 7 shows a 4^(th)-order channel add-drop filter design 70 having aninput waveguide 75, an output waveguide 76, and a system of four coupledmicroring resonators 71-74 of substantially equal radius. In theembodiment shown, the system of four coupled microring resonators 71-74forms a single coupling loop, which has a geometrical tilt of about⅛^(th) of the microring guided wavelength to provide a negative LCC,i.e., 180° LCP. The ring radii need not, however, be similar. Inparticular, the first cavity 71 is coupled to the input waveguide 75 andto the second cavity 72. The second cavity 72 is in turn coupled to thethird cavity 73, which is in turn coupled to the fourth and final cavity74. The fourth cavity 74 is also coupled to the output waveguide 76 andto the first cavity 71. FIG. 8 shows the same structure 70 in asquare-grid configuration, but without a geometrical tilt, and thus thestructure 70 has a positive LCC, i.e., 0° LCP. In the embodiments shownin FIGS. 7 and 8, an even number of microring resonators 71-74 (i.e.,four) form the coupling loop so that light will only propagate in asingle direction in each of the resonators 71-74, when an optical inputsignal is sent into only one of the input ports. While only microringresonators 71-74 are shown in the embodiments depicted in FIGS. 7 and 8,the structure 70 (and any structure described herein as employingmicroring resonators) may more generally include any type of ring-typeresonator 71-74, such as a racetrack resonator. Most generally, weconsider a ring-type resonator as any structure consisting of a lengthof waveguide whose beginning and end are joined so as to form a closedloop path for propagating light; which we also refer to as atraveling-wave resonator.

In one embodiment, the structures 70 in FIGS. 7 and 8 have anon-all-pole (i.e., a pole-zero) spectral response from the input port77 to drop port 78 that may not be obtained in SCC structures, such asthe SCC structure 310 of FIG. 1. Thus, with the structures 70 in FIGS. 7and 8 supporting pole-zero input-to-drop responses, by engineering thepositions of transmission nulls, quasi-elliptic filter responses may berealized. The structures 70 in FIGS. 7 and 8 have 4 system poles(resonances) due to the four resonant cavities 71-74 (specifically eachpole corresponding to one of the four supermodes of the coupled systemof the four resonant cavities, with ports), but introduce only twofinite-frequency zeros. The existence of two transmission zeros can beunderstood by thinking of the filter 70, in the case of off-resonantexcitation, as a Mach-Zehnder (MZ) interferometer formed of resonators.The MZ has two feed-forward paths from the input port 77 to drop port78, which interfere, the first being (input port 77)-(cavity 71)-(cavity74)-(drop port 78), and the second being (input port 77)-(cavity71)-(cavity 72)-(cavity 73)-(cavity 74)-(drop port 78). In general, anexcited cavity couples energy in all directions (i.e. out to alladjacent cavities coupled to it, regardless from which direction asignal is coming), so a feed-forward (interferometer-like) model cannotbe assumed for the flow of energy. However, in the off-resonant regime,the energy coupled from one cavity to the next in the forward direction(e.g., cavity 71 to 72) is small, and that returning in the backwarddirection (e.g., cavity 72 to 71) is the square of this smallcoupling—i.e., it is even smaller. Thus an MZ feed-forwardinterferometer of cavities is a reasonable asymptotic model faroff-resonance, but turns out to hold valid with reasonable accuracy evenjust outside the resonant passband (i.e. fairly near resonance). Zerosare of importance for passband engineering in particular when placedoutside the passband at real or complex frequency detunings, so thismodel is relevant.

Then, in this embodiment, the number of transmission zeros introducedinto the input 77-to-drop-port 78 spectral response is equal to thenumber of cavities bypassed by the shortest path from input to output.Since cavities 72 and 73 are bypassed by the coupling from cavity 71 tocavity 74, the filters 70 depicted in FIGS. 7 and 8 have 2 transmissionzeros in the drop response, and the filter rolloff is 4^(th)-order(approximately 80 dB/decade of frequency detuning) near the passband asillustrated by the long-dash line 104 in FIG. 10, but reduces to2^(nd)-order (approximately 40 dB/decade) at detunings from centerwavelength larger than those of the transmission zeros, as illustratedby the dotted line 103 in FIG. 10. This is because far from resonanceonly a collection of 4 poles and 2 zeros near a common center wavelengthare seen, and hence there is apparent cancellation of 2 poles by 2zeros, leaving 2 poles to effect the remaining rolloff of the passband.

FIG. 9 shows a structure that illustrates a general MZ model 90 of aloop coupling (and is itself a valid loop-coupled structure) with N+4cavities 92, where N of the cavities along an energy propagation path 96are bypassed by a second interferometric path 94, thus creating Ntransmission zeros.

FIG. 10 a shows the actual response (input port 77-to-drop port 78),including the effect of transmission zeros, of the device 70 shown inFIG. 7. More specifically, FIG. 10 a shows a frequency-normalizedperformance of the device 70 of FIG. 7, having an approximately ⅛^(th)wavelength tilt. The horizontal axis shows normalized frequency units inrad/s. An approximately flat-top response with two real-frequency zeroson the sides of the passband that help to increase the sharpness of thepassband is illustrated. Such a sharper passband means thatwavelength-division-multiplexed (WDM) channels may be more denselypacked into the available optical spectrum, while being able to maintaina similar out-of-band rejection ratio, number of resonators, andinput-to-drop loss. The filter 70 whose performance is illustrated inFIG. 10 a by line 101 uses ECCs: {2/τ_(in), μ_(71,72), μ_(72,73),μ_(73,74), 2/τ_(out), μ_(74,71)}={2.2, 1, 1, 1, 2.2, −0.2} rad/s. Thenegative sign in the last coupling indicates a negative product of ECCs,i.e., a negative LCC.

With regard to coupling coefficients used to describe the devices of thepresent invention, two descriptions are used: energy couplingcoefficients (ECCs), and power coupling coefficients (PCCs). Each may beconverted to the other. ECCs are representative of a general concept,whose definition is well known in the framework of coupled mode theoryin time, used to describe the coupling: (a) between a first and a secondresonant cavity, or (b) between a resonant cavity and a waveguide oroptical access port. Coupling between a resonant cavity mode and awaveguide or port is described by an energy-amplitude decay rate,τ_(n)≡1/τ_(n) of the energy amplitude of the resonant mode into thewaveguide or port, where n is used to index an arbitrary couplingcoefficient, r_(n) is the decay rate, and τ_(n) is the decay time. Theenergy is the square of the magnitude of the energy amplitude, so theenergy decays with energy decay rate 2/τ_(n), with units equivalent torad/s. On the other hand, coupling between a first and a second resonantcavity mode is described by the rate at which energy passes back andforth between the coupled cavities, measured as a frequency in rad/s andindicated by a complex-number labeled μ_(i,j), referred to elsewhere asthe ECC μ_(i,j), whose magnitude is the coupling frequency and whosephase indicates the location of coupling with respect to a referencepoint chosen for the resonant mode. The phase of the complex-number ECCμ_(i,j) plays a role in the LCP. For all structures described herein,all cavity-waveguide ECCs are specified as energy decay rates 2/τ_(n),and all cavity-cavity ECCs are specified as coupling rates μ_(i,j), bothin consistent rad/s units unless stated otherwise. Moreover, all ECCsare typically collected and given as a set. For example, for afour-cavity SCC structure, the ECCs are listed from input to output:{2/τ_(in), μ₁₂, μ₂₃, μ₃₄, 2/τ_(out)}. For more complex (e.g.,loop-coupled) structures, the sequence in which coefficients are listedis explicitly stated. Furthermore, a set of “normalized ECCs” may bestated, which hold for a “prototype filter” whose passband cutofffrequency is 1 rad/s from the center frequency (i.e., full passbandwidth of 2 rad/s), as is common practice in electrical circuit design.The actual ECCs may be obtained by unnormalizing, i.e., by multiplyingthe normalized ECCs by the desired actual half-bandwidth of the filter,in rad/s.

When using traveling-wave cavities, including microring and racetrackresonators, the cavity-cavity and cavity-waveguide coupling aretypically achieved by evanescent coupling across directional couplerregions. Directional couplers are described by a PCC, which determinesthe fraction of power incident into the input port of one waveguide thatis coupled to the cross-port, i.e., to the output port in the otherwaveguide. In this case, the ECCs of a resonant structure can betranslated to PCCs corresponding to the directional couplers in thestructure. The conversion is done in two steps: ECC to PCC conversion,and PCC correction for finite FSR. We note, as known in the literature,that for purposes of the coupling coefficient formulas the FSR isdefined not as the actual frequency spacing between two adjacentresonant orders, but rather as FSR≡v_(group)/L_(roundtrip), where thegroup velocity, v_(group), is evaluated at the resonance frequency ofinterest and L_(roundtrip) is the round-trip length of the resonantcavity. If the group velocity is given in meters per second and theround-trip cavity length in meters, then the FSR is given in cycles persecond (Hertz). In the first step of the conversion, forcavity-waveguide coupling, the PCC κ_(n) ² (where κ_(n) is amplitudecoupling), is given by

κ_(n) ² FSR _(m)=2/τ_(n),  (1)

where FSR_(m) is the FSR of the particular cavity in question, in unitsof Hz (cycles/second) if the energy decay rate 2/τ_(n), is given inrad/s. For cavity-cavity coupling, the power coupling between resonantmode i of a first cavity and resonant mode j of a second cavity is givenby

κ_(i,j) ² FSR ₁ FSR ₂=|μ_(i,j)|²,  (2)

where FSR₁ and FSR₂ are the FSRs of the respective modes of the firstand second cavity, in Hz, if μ_(i,j) is in rad/s. Note that the secondcavity may, for certain couplings, be the same cavity as the firstcavity, in which case the associated ECC μ_(i,j) describes coupling oftwo resonant modes in one cavity. Furthermore, for j=i, μ_(i,i)represents coupling of a mode to itself due to a perturbation. Thisnomenclature for coupling coefficients and decay rate variables, and theassociated relationships, are well described in literature.

Once the PCCs are obtained from the ECCs, a second mapping, also knownin the art, is applied to account for the finite FSR of the cavities.Each PCC is replaced by a scaled version of itself as:

$\begin{matrix}\left. \kappa^{2}\leftarrow\frac{\kappa^{2}}{\left( {1 + \frac{\kappa^{2}}{4}} \right)^{2}} \right. & (3)\end{matrix}$

In the obtained PCC set, all numbers are unitless power couplingfractions, one per directional coupler.

Referring again to FIG. 10 a, which illustrates the physicalimplications of loop coupling on the transmission zeros and spectralresponse, the asymptotic Mach-Zehnder model described previously usingthe structure 90 depicted in FIG. 9, and where the model treats thedevice 70 depicted in FIG. 7 as a two-path feed-forward interferometerand is valid off-resonance, is also shown by medium-dashed line 102. Thegood agreement between the asymptotic model line 102 and the exactresponse line 101 outside the passband, and even for the locations ofthe transmission zeros that are placed quite near the passband,validates the use of this physical model to understand the physicaloperation of loop-coupled resonant devices. Two asymptotes on FIG. 10 a(lines 103 and 104) show that far from resonance the filter rolloff issecond-order (line 103), while near the passband it is sharper, closerto fourth-order rolloff (line 104).

FIG. 10 b shows the location in the complex-frequency plane (with originplaced at the center of the passband, thus showing complex frequencydetuning) of the transmission zeros for several exemplary LCPs. For thestructure 70 depicted in FIG. 7, with 180° LCP, the zeros 105 arelocated diametrically on opposite sides of the center wavelength, on thereal-frequency axis (i.e., the horizontal axis). The detuning distanceof the zeros 105 from the passband center is inversely proportional tothe magnitude of the coupling coefficient between cavities 71 and 74,μ_(74,71) and directly proportional to the other three cavity-cavitycouplings, μ_(72,71), μ_(73,72) and μ_(74,73). When μ_(74,71)=0, thezeros 105 go to arbitrarily large detuning and the filter 70 reverts toa standard SCC all-pole design. For the structure 70 depicted in FIG. 8,when the LCP=0, the zeros 106 are located at imaginary frequencydetunings, again on diametrically opposite sides of the passband center.By varying the LCP, for example by tilting the structure 70 of FIG. 8toward that of FIG. 7, the pair of zeros 106 may be rotated to anylocation on the locus circle 107 shown in FIG. 10 b, where the radius isdetermined approximately (from the asymptotic MZ model derived using thestructure 90 in FIG. 9) by the product of the couplings along the longcavity-interferometer path 96 divided by the coupling along the shortpath 94. The square root of this quotient of couplings is the detuningfrom center frequency, and the phase of the detuning is related directlyto the direction in the complex frequency plane along which the zerosare placed—which can be shown to depend only on the LCP, Φ, as indicatedby equation (4), below:

$\begin{matrix}{{\delta\omega} \approx {{\pm \sqrt{\frac{\mu_{74,73}\mu_{73,72}\mu_{72,71}}{\mu_{74,71}}}}^{{- {j{({\Phi + \pi})}}}/2}}} & (4)\end{matrix}$

Note that equation (4) holds only for 4-cavity coupling loops, such asthose shown in FIGS. 7 and 8, with a waveguide attached to each of twoadjacent cavities; however, similar expressions may be derived for anyorder coupling loops.

It is of interest to create filters with a full N poles and N zeros (perFSR) with controllable frequency positions, for an N-cavity system,because the sharpest achievable spectral response for bandpass filtersis known to be the elliptic function response, which requires N polesand N zeros. Realizing this response function is optimal in the sense ofachieving the sharpest filter rolloff with a given number of cavities,i.e., a given order of the resonant system. The previous discussionmakes clear that a filter with an equal number of poles and zeros (e.g.,four) will have no rolloff at large enough detuning, because all thepoles and zeros cancel when observing the passband wavelength from thevantage point of a large wavelength detuning. This is consistent withthe spectral shape of an elliptic filter response function, whichreaches a constant level at large detuning.

Referring now to FIG. 11, one way to obtain a wavelength-independenttransmission at a given rejection level is to introduce a direct, weakcoupling 115 between an input waveguide 116 and an output (drop-port)waveguide 117, in front of or behind the position at which the system ofcoupled cavities 111-114 is placed between the two waveguides 116, 117.A very weak directional coupler 115 with 0.001 power coupling fraction(−30 dB) may be used for an elliptic filter demanding a 30 dBout-of-band rejection in the drop port 118. In one embodiment, thelength of the input waveguide 116 between the directional coupler 115and the point at which the microring resonator 111 couples to the inputwaveguide 116 is substantially equal to the length of the outputwaveguide 117 between the directional coupler 115 and the point at whichthe microring resonator 114 couples to the output waveguide 117. Inanother embodiment, a phase shift 119 (here of 180°) is introduced inthe input waveguide 116 between the coupler 115 and the filter(represented by cavities 111-114) in order to produce real-frequencyzeros outside the passband in the drop port 118 spectrum (as illustratedby the solid-curve 123 spectrum in FIG. 12). Hence, the (typically weak)coupling of the input waveguide 116 and the transmission (drop-port)waveguide 117 of interest is an alternative and additional way ofcreating controllable transmission zeros. It should be noted that thisaddition does not harm the robustness (e.g., to fabrication variations)of the design because the coupling is very weak, and any error in thecoupling coefficient in the directional coupler 115 only slightlychanges the rejection level. By contrast, all-pass decomposition (AD)filters, such as structure 330 illustrated in FIG. 3, that rely on 3 dBcouplers are highly sensitive to small errors in the 3 dB splittingratio, which may lead to poor rejection levels over part or all of theoperating wavelength range when either fabrication errors areintroduced, or an insufficiently wavelength-independent coupler isdesigned.

In one embodiment, the optical coupling between the input waveguide 116and the output waveguide 117 introduced by the directional coupler 115is substantially broadband over several passband widths, for exampleover at least three passband widths or over at least ten passbandwidths. It should also be noted that the phase shift 119 may beintroduced in the output waveguide 117, as opposed to in the inputwaveguide 116, between the directional coupler 115 and the four cavities111-114. Moreover, although the structure 110 is shown in FIG. 11 toinclude four microring resonators 111-114 coupling the input waveguide116 to the output waveguide 117, more generally any number of resonatorsmay be employed, including a single resonator, or a plurality resonatorsthat are loop-coupled or that are not loop-coupled. As shown in FIG. 11,the four microring resonators 111-114 may form a coupling loop, asdescribed above for the structure 70 of FIG. 7.

The structures 70 and 110 in FIGS. 7 and 11, respectively, may be usedin flat-top channel add-drop filters having higher selectivity than SCCall-pole designs. A comparison of exemplary 4^(th)-order filter designsusing the three geometries (i.e., structure 70 depicted in FIG. 7,structure 110 depicted in FIG. 11, and a 4^(th)-order SCC all-poledesign similar to the 3^(rd)-order structure 310 in FIG. 1) is shown inFIG. 12. In one embodiment, all three filters are designed to have a 30dB in-band extinction in the input-to-through-port response across abouta 40 GHz passband so that their out-of-band rolloff in the drop port maybe compared. The presence of two zeros in the loop-coupled design 70 ofFIG. 7 makes its response (dashed line 121 in FIG. 12) much sharper thanthat of the standard SCC structure with a plain Chebyshev response(dotted line 122 in FIG. 12). Using the structure 110 of FIG. 11, withabout −30 dB direct coupling of the input and output waveguides 116, 117introducing another 2 real-frequency transmission zeros, gives thesharpest, elliptic-function-shaped response (solid line 123 in FIG. 12).In practice, for an N^(th) order structure, it may be preferable todesign for N−1, N−2, or fewer zeros so that the drop-port response rollsoff at a rate of at least first or second order at large detuning.

The design parameters of the filter designs shown in FIG. 12 may, invarious embodiments, be as follows. For the standard SCC design 124(shown in the top left-hand corner of FIG. 12) having cavities 125, 126,127, 128, a Chebyshev filter may be designed with normalized ECCs:{2/τ_(in), μ_(125,126), μ_(126,127), μ_(127,128), 2/τ_(out),μ_(128,125)}={3.078, 1.154, 0.8349, 1.154, 3.078, 0} rad/s. For theshown 40 GHz-bandwidth filters of the standard SCC design 124, thecorresponding actual ECCs may be {386.8, 145, 104.9, 145, 386.8, 0}Grad/s (where Grad=gigaradian=10⁹ radians). For a microring-resonatorimplementation of the standard SCC design 124, using, in one embodiment,microrings with a 3 THz FSR that is typically achievable inhigh-index-contrast microring resonators, the corresponding PCCs may be{κ_(in) ², κ_(125,126) ², κ_(126,127) ², κ_(127,128) ², κ_(out) ²,κ_(128,125) ²}={0.121, 0.002335, 0.001222, 0.002335, 0.121, 0}. Asbefore, the coupling coefficients subscripted with pairs of numberslabel and indicate cavity-cavity coupling, while ‘in’ and ‘out’ indicatecoupling to the input and output waveguides of the standard SCC design124.

For the loop-coupled structure 70 of FIG. 7, which may provide aquasi-elliptic response, the normalized ECCs may be: {2/τ_(in),μ_(71,72), μ_(72,73), μ_(73,74), 2/τ_(out), μ_(74,71)}={3.018, 1.098,0.9036, 1.098, 3.018, −0.2287} rad/s. For the shown 40 GHz-bandwidthfilters of the loop-coupled structure 70, the corresponding actual ECCsmay be {379.3, 137.9, 113.6, 137.9, 379.3, −28.73} Grad/s. The productof the four cavity-cavity couplings is negative, indicating a LCP of180° in the loop-coupled structure 70, which leads to the tworeal-frequency-axis zeros. For a microring implementation of theloop-coupled structure 70 with a 3 THz FSR, the corresponding PCCs maybe {κ_(in) ², κ_(71,72) ², κ_(72,73) ², κ_(73,74) ², κ_(out) ²,κ_(74,71) ²}={0.1188, 0.002111, 0.001432, 0.002111, 0.1188, 0.0000917}.

For the loop-coupled structure 110 of FIG. 11 with direct couplingbetween the input and output waveguides 116, 117, which may provide anelliptic response with 4 nulls, the normalized ECCs may be: {2/τ_(in),μ_(111,112), μ_(112,113), μ_(113,114), 2/τ_(out), μ_(114,111)}={2.96,1.076, 0.919, 1.076, 2.96, −0.3274} rad/s. For the shown 40GHz-bandwidth filters of the loop-coupled structure 110, thecorresponding actual ECCs may be {372, 135.3, 115.5, 135.3, 372, −41.14}Grad/s. The product of the four cavity-cavity couplings is negative,indicating a LCP of 180° in the loop-coupled structure 110, whichcontributes to two of the four real-frequency-axis zeros. In oneembodiment, the input and output waveguides 116, 117 are coupled in adirectional coupler 115, with 0.00118 power coupling, located before thecavities 111-114. In addition, the optical paths from the symmetricdirectional coupler 115 to cavities 111 and 114, respectively, may bemade to differ by 180° by the introduction of a phase shift 119 to pathsof otherwise identical lengths, each being in one of the waveguides 116,117. This may be done by adding a half-guided-wavelength of extra lengthof waveguide in one of the two arms 116, 117, at the filter centerwavelength, or by using a thermooptic phase shifter, or by other means.For a microring implementation of the loop-coupled structure with a 3THz FSR, the corresponding PCCs may be {κ_(in) ², κ_(111,112) ²,κ_(112,113) ², κ_(113,114) ², κ_(out) ², κ_(114,111) ²}={0.1167,0.002031, 0.001481, 0.002031, 0.1167, 0.0001881}.

The described 4^(th)-order structures with a single coupling loop may berealized in microring resonator technology in a single lithographiclayer. By using multiple loop couplings, the control of multiple zerosof transmission may be obtained, and high-performance filters andoptical delay lines that are non-minimum-phase and thus are notconstrained in amplitude and phase response by the Kramers-Kronigcondition may be designed.

FIGS. 13 a and 13 b each illustrate one embodiment of another structureaccording to the present invention. The structure 130 includes two rowsof coupled resonators 132, with equal number of resonators 132 in eachrow and with each resonator 132 coupled both horizontally andvertically. In one embodiment, the inter-row (i.e., vertical) couplings(for example, the inter-row energy coupling coefficients) are weakerthan the intra-row (i.e., horizontal) couplings. More specifically, FIG.13 a shows a realization of an exemplary 14^(th)-order structure 130 inmicroring-resonator technology, while FIG. 13 b shows a schematic of anexemplary, functionally equivalent structure 130 realized usingstanding-wave cavities 132 (e.g., photonic crystal microcavities), eachcavity 132 having a two-lobe resonant mode. In FIG. 13 b it isunderstood that horizontally and vertically adjacent cavities 132 may besubstantially coupled in the schematic, but not those diagonallyadjacent. In one embodiment, the structures 130 use positive andnegative LCCs. The structures 130 have six independent coupling loops(others are redundant), which determine the spectral features (anadditional direct phase from input to output only sets the overallphase, which has no consequence for the spectral shape).

The structures 130 illustrated in FIGS. 13 a and 13 b are capable ofrealizing non-minimum-phase responses, in particular filter responseshaving a flat-top passband and linear phase simultaneously, and withoutthe need for additional all-pass dispersion compensators following thestructure, as is necessary to linearize the phase of SCC structures. Thestructure 130 is optimal in the sense that it supports the mostselective (quasi-elliptic) amplitude responses for a given resonanceorder. In the structure 330 of FIG. 3, an architecture known in the artto allow elliptic optical filters, the extinction ratio is sensitive toobtaining a balanced, broadband 3 dB splitter. By contrast, thestructures 130 of FIGS. 13 a and 13 b are very robust, and at onceprovide highly compact implementations of these optimal filterresponses. Having distributed dispersion compensation within thestructure 130 has advantages not only for filters, but also fornonlinear applications where distributed dispersion management ispreferable.

The structure 130 may directly implement an “equidistant linear-phasepolynomial” response function that is known in the art. This responsefunction may simultaneously provide optimally flattened amplitude andgroup-delay responses (i.e., linearized phase). This is generally donewith fewer resonators (e.g., the minimum) than required in an equivalentcascade of an all-pole filter (e.g., an SCC) followed by an all-passdispersion compensator. A second use of the device 130 may be forslow-light optical delay lines. Coupled resonator waveguides (CRWs),i.e., SCC structures, with equal coupling coefficients have largeamplitude oscillations and dispersion in the resonant passband, whileSCC filters with a properly apodized coupling coefficient distributionhave near-unity transmission in the passband, but both retain largedispersion due to the Kramers-Kronig constraint. It has previously beenshown (for example in J. B. Khurgin, “Expanding the bandwidth ofslow-light photonic devices based on coupled resonators,” Mar. 1,2005/Vol. 30, No. 5/OPTICS LETTERS, pp. 513) that adding individualside-coupled cavities to a CRW can cancel the lowest-order dispersionterm in the input-to-drop-port response. The present structure 130provides optimally flattened dispersion to high order in the sense ofthe equidistant linear-phase polynomial, and uses no extra resonators.Along with linearized phase, the structure 130 also simultaneouslyprovides a unity amplitude transmission in the passband unlikelinear-phase Bessel filters. Bessel filters are the best that anall-pole filter can do in terms of achieving linear phase in thepassband, but they necessarily sacrifice amplitude response by having arounded passband (rather than flat).

The design of a flat-top, linear-phase, high-order (30 cavities)resonant structure 134, depicted in FIG. 13 c, is now described asanother exemplary embodiment. An electrical circuit that gives aflat-top, linear phase response and gives the realization of itsresponse function in the optical structure 134 depicted in FIG. 13 c, byproviding a mapping of the circuit's parameters to the relevantparameters of the optical structure 134, is also now described. To theseends, FIG. 14 a shows one embodiment of an electric circuit prototype140 that supports response functions with simultaneously optimallyflattened passband amplitude and group delay in transmission, as derivedfrom an “equidistant linear-phase polynomial” given in the literature.This electrical circuit 140 is topologically compatible with the opticalcoupled-cavity structure 134 depicted in FIG. 13 c, each capacitor 142being equivalent to a resonant cavity 136 of the structure 134. Acorrespondence exists between the coupled-mode theory in time (CMT)model of the optical filter 134 and the circuit equations of thelow-pass prototype circuit 140 with “resonators” at zero resonantfrequency (i.e., sole capacitors, which may be imagined to have aninfinite inductance in parallel—an open circuit, such that the resonantfrequency is zero, 1/√{square root over (LC)}=0).

A mapping between electrical resonators and couple mode parameters canbe found by physical arguments. The capacitors 142 may be considered asresonators at zero frequency, that is, having in parallel an inductor ofinfinite inductance, i.e., an open circuit, such that the resonancefrequency ω_(o)≡1/√{square root over (LC)}=0. By noting that themagnitudes of the CMT energy coupling coefficients, μ_(i,j), are realnumbers in units of rad/s and that these real numbers represent thefrequency of energy exchange between two energy storage elements, thecircuit 140 may be mapped to the CMT model 134 by inspection byconsidering each part of the circuit 140 separately. FIG. 14 b showsthat a pair of capacitors 142 (each representing a resonator) connectedby an immitance (impedance or admittance) inverter 144 is equivalent toan L-C resonator 146. The resonance frequency is the frequency of energyexchange between the two capacitors, i.e. resonators. In addition,referring again to FIG. 14 a, an input port 141 is coupled to an input“resonator” 142 (C₁) and an output port 143 is coupled to an output“resonator” 142 (C₁). The decay rate of the input and output“resonators” 142 (C₁) is an R-C time constant formed by the capacitanceand the characteristic impedance of the port (which may, without loss ofgenerality, be assigned as Z_(o)=1Ω, as illustrated in FIG. 14 a). Aprototype circuit 140 of the type shown in FIG. 14 a may be obtained,for various filter responses of interest, from known literature, thusproviding capacitor (C_(n)) and immitance inverter (K_(n)) values forthe structure. The CMT parameters of the optical structure 134 that isequivalent to the circuit 140 depicted in FIG. 14 are given by:

$\begin{matrix}{{\mu_{n,{n + 1}}^{2} = {\mu_{{N - n},{N - n + 1}}^{2} = \frac{1}{C_{n}C_{n + 1}}}}{{1/\tau_{in}} = {{1/\tau_{out}} = \frac{1}{C_{1}}}}{\mu_{n,{N - n + 1}} = \frac{K_{n}}{C_{n}}}} & (5)\end{matrix}$

for n=1 . . . N/2, where N is the filter order (number of cavities),with coupling coefficient indices in equation (5) referring to thenumbering of the cavities 136 in the structure 134 of FIG. 13 c.

The graph of FIG. 15 a compares the optical structure 134 depicted inFIG. 13 c with a typical maximally-flat SCC filter (see FIG. 1), whereboth include 30 rings labeled 1, 2, 3, . . . 30 (i.e., are of orderN=30). In one embodiment, both the optical structure 134 and the SCCfilter are designed to have a 40 GHz 3 dB bandwidth. In such a case, thenormalized ECCs for the SCC filter may be: {2/τ_(in), μ_(1,2), μ_(2,3),μ_(3,4), . . . , μ_(29,30), 2/τ_(out)}={19.11, 30.54, 6.176, 2.696,1.537, 1.011, 0.7295, 0.5619, 0.455, 0.3836, 0.3346, 0.3006, 0.2773,0.2621, 0.2535, 0.2507, 0.2535, 0.2621, 0.2773, 0.3006, 0.3346, 0.3836,0.455, 0.5619, 0.7295, 1.011, 1.537, 2.696, 6.176, 30.54, 19.11} rad/s,while the actual ECCs for a 40 GHz-wide design may be: {2401, 3838,776.1, 338.8, 193.1, 127.1, 91.67, 70.61, 57.18, 48.21, 42.05, 37.77,34.84, 32.93, 31.86, 31.51, 31.86, 32.93, 34.84, 37.77, 42.05, 48.21,57.18, 70.61, 91.67, 127.1, 193.1, 338.8, 776.1, 3838, 2401} Grad/s.

The normalized ECCs (for a 2 rad/s full passband width) for the filterprototype used for the loop-coupled structure 134 of FIG. 13 c may be,for the top row of couplings, left to right: {2/τ_(in), μ_(1,2),μ_(2,3), μ_(3,4), . . . , μ_(14,15)}={11.13, 3.241, 1.489, 1.016,0.8002, 0.6808, 0.6083, 0.5623, 0.5325, 0.5125, 0.4949, 0.4675, 0.4201,0.3503, 0.2521} rad/s; and for the vertical couplings, left to right:{μ_(1,30), μ_(2,29), μ_(3,28), . . . , μ_(15,16)}={−0.006747, −0.006618,−0.00631, −0.005695, −0.00448, −0.00204, 0.002908, 0.0127, 0.03041,0.05671, 0.0831, 0.09419, 0.08752, 0.07632, 0.06901} rad/s. The actualECCs for a 40 GHz bandwidth may be: for the top row, left to right:{1398.2, 407.3, 187.1, 127.7, 100.6, 85.55, 76.45, 70.66, 66.91, 64.41,62.19, 58.75, 52.79, 44.03, 31.68}; for the vertical couplings, left toright: {−0.8478, −0.8316, −0.793, −0.7156, −0.5629, −0.2563, 0.3654,1.595, 3.822, 7.126, 10.44, 11.84, 11, 9.591, 8.672}; and for the bottomrow, the same as for the top row. Without loss of generality, in FIG. 13c all the top and bottom row ECCs, as well as the rightmost verticalcoupling, are arbitrarily assigned as positive real. The signs of theremaining vertical couplings are directly related to the independentLCPs of the structure 134. This convention is used to permit mapping ofthe circuit 140 theory results directly to the optical structure 134. Inorder to accommodate the change of sign of the ECCs, there is a singlecoupling geometry kink 137 near the middle of the loop-coupled structure134, representing an approximately ⅛^(th) guided wavelength tilt. Byconsidering the LCP of each set of four adjacent rings 136 (two in thetop row and two in the bottom row), it can be seen that all LCPs arezero, except for the coupling loop consisting of cavities 6, 7, 24, and25, whose LCP is 180° due to the presence of the kink 137 in thestructure 134 between the 6^(th) and 7^(th) cavity. In one embodiment,each coupling loop (other than the coupling loop consisting of cavities6, 7, 24, and 25), in which all the rings in the coupling loop haveequal radii, has a symmetric near-square (or, in the case where thecoupling gaps are variable, trapezoidal) geometry in order to keep theLCPs of those coupling loops substantially equal to zero.

With reference again to FIG. 15 a, the resulting spectra of thestructure 134 depicted in FIG. 13 c and of the typical maximally-flatSCC filter show that the passband flatness and near-passband rolloff ofthe two filters is the same. However, the loop-coupled structure 134,which uses the prototype circuit for equidistant linear phase filters,sacrifices rolloff beyond 30 dB extinction, in this case, to placefinite transmission zeros (at complex-frequency detuning) that allowoptimally flattened group delay, i.e., suppressed dispersion. The groupdelay response spectra of the structure 134 depicted in FIG. 13 c and ofthe typical maximally-flat SCC filter are shown in FIG. 15 b and theirdispersion response spectra are shown in FIG. 15 c. As illustrated, thedifference between the two structures is very substantial. For example,the dispersion is less than 20 ps/nm (the typical telecom requirementfor 10-40 Gb/s signals) over only 6% of the bandwidth in the standardSCC design, and over 80% of the bandwidth in the structure 134 depictedin FIG. 13 c. Furthermore, the dispersion is less than 0.05 ps/nm over75% of the passband width in the loop-coupled structure 134. A 25 pspulse passing through the 40 GHz passband is delayed by about 10 pulsewidths, with no substantial dispersion or pulse distortion.

For a chosen exemplary cavity FSR of 2 THz, typical for a microringresonator, the ECCs given above are translated to power couplingcoefficients for the directional couplers. In such an embodiment, thefinal power coupling coefficients for the loop-coupled structure 134depicted in FIG. 13 c with a 40 GHz bandwidth and a 2 THz microringresonator FSR are as follows.

Referring to FIG. 13 c, the top row of couplings is (left to right):{κ_(in) ², κ₁₂ ², κ₂₃ ², . . . , κ_(14,15)}={0.5066, 0.04063, 0.00871,0.00407, 0.002524, 0.001828, 0.00146, 0.001247, 0.001119, 0.001036,0.0009663, 0.0008626, 0.0006965, 0.0004844, 0.0002509}. Referring toFIG. 13 c, the vertical couplings, left to right, are: {κ_(1,30) ²,κ_(2,29) ², κ_(3,28) ², . . . , κ_(15,16) ²}={1.797e-007, 1.729e-007,1.572e-007, 1.28e-007, 7.922e-008, 1.642e-008, 3.339e-008, 6.363e-007,3.652e-006, 1.269e-005, 2.726e-005, 3.503e-005, 3.024e-005, 2.299e-005,1.88e-005}. The bottom row of couplings is the same as the top row ofcouplings.

In one embodiment, the final power coupling coefficients for the typicalmaximally-flat SCC filter with a 40 GHz bandwidth and 2 THz FSR are,from input to output: {κ_(in) ², κ₁₂ ², κ₂₃ ², . . . , κ_(29,30) ²,κ_(out) ²}={0.7103, 0.1136, 0.02409, 0.01059, 0.006049, 0.003984,0.002876, 0.002216, 0.001795, 0.001513, 0.00132, 0.001186, 0.001094,0.001034, 0.001, 0.0009893, 0.001, 0.001034, 0.001094, 0.001186,0.00132, 0.001513, 0.001795, 0.002216, 0.002876, 0.003984, 0.006049,0.01059, 0.02409, 0.1136, 0.7103}. In this case, for example, theleading power coupling coefficient was about 1.2 prior to finite-FSRcorrection, i.e., above 100%. However, the proper FSR scaling usingequation 3, set forth above, provides 0.71 as the correct coupling, andthe resulting set of power coupling coefficients may be verified bysimulation to produce the desired flat-top passband.

In general, the ECCs, as listed in the previous examples, scale linearlywith passband width, so they are scaled to obtain different bandwidths.Equations (1) and (2), set forth above, may be applied to resonatorshaving nonidentical FSRs. In this way, the same design may be scaled tovarious physical structures.

The specific examples given so far have been used to illustrate theutility of loop coupling, and of the engineering of the LCP(s). Moregenerally, coupled-cavity structures that include an input waveguide andat least one output waveguide, and that are connected by acoupled-resonator system with non-trivial coupling loops, will introducefinite, complex-frequency transmission zeros into the input-to-dropresponse spectrum. Exemplary embodiments of such generic structures areillustrated in FIG. 16 using standing-wave cavities and in FIG. 17 ausing microring resonators.

In the structure 160 depicted in FIG. 16, four coupling loops 161-164are shown. With reference to coupling loop 161, the LCPΦ₁=argument(μ_(2,3)·μ_(3,4)·μ_(4,5)·μ_(5,6)·μ_(6,7)·μ_(7,2)) is thephase of the product of six ECCs. For the standing-wave-resonatorstructures of FIG. 16, ports 165-167 include direct-coupled waveguides,in which there is an incoming (input-port) signal and an outgoing(through-port) signal as the forward and backward wave in the waveguide.It is assumed that the end of the waveguide is substantially reflectiveso that when there is no substantial energy in the input resonator, asubstantial fraction of the input power is reflected, preferably all. Analternative, low-loss way to excite a single high-Q standing-wave cavityto accomplish this is by using a symmetric waveguide loop mirror 176,illustrated in FIG. 17 b. As illustrated in FIG. 17 c, by evanescentlycoupling the end 177 of the loop mirror 176, which has a standing waveexcited, to a standing-wave cavity 178, and placing the cavity 178 suchthat the cavity mode and loop mirror standing wave pattern are notorthogonal, an efficient mechanism for exciting standing-wave structuresis provided.

FIG. 17 a depicts one embodiment of a generic microring-resonatorimplementation of loop-coupled resonators. The non-trivial couplingloops 171-174 may each contain an even number of resonators 175, so asto avoid coupling to the degenerate, contradirectional modes (i.e., oneresonance excited per cavity), when the system is excited by a signalentering a single input port. For the coupling loop 171, the LCP may benon-zero, e.g., 180°, if the tilt is approximately ⅛^(th) of the guidedwavelength. In the case where the four coupling coefficient magnitudesin the coupling loop 171 are non-identical and employ differentevanescent coupling gap widths, the geometry tilt angle required will bechanged from ⅛^(th) wavelength and configured so as to still maintain a180° LCP. If the coupling loop 171 is square and has approximately equalPCC magnitudes, the LCP is 0°. For coupling loops comprising a largernumber of cavities, there is a larger number of adjustable degrees offreedom and the tilt angles may be chosen accordingly to provide a givenLCP, as described in general.

An exemplary physical realization of the 4^(th)-order loop-coupledstructure 70 of FIG. 7 is depicted in FIGS. 18 a-c, while an exemplaryphysical realization of a 4^(th)-order standard SCC filter is depictedin FIGS. 19 a-c. Both may be realized using a silicon-rich siliconnitride (refractive index n=2.2) ring waveguide core with a 900×400 nmcross-section, with silica (n=1.45) undercladding and air cladding onthe top and sides of the core. An overetch of 100 nm (beyond the 400 nmcore thickness) may be used when patterning the waveguides. All otherplanar dimensions and coordinates that may be used to realize the4^(th)-order loop-coupled structure 70 of FIG. 7 in the SiN materialsystem are contained in FIG. 18 b. For a zero LCP, the centers 181 ofthe rings 182 of the structure 70 depicted in FIG. 18 c may be arrangedon the vertices of a trapezoid (e.g., a rectangle) with a horizontalaxis of symmetry. In the design in FIGS. 18 a-c, the angles show a smalltilt (of about 0.86°) in order to set a negative LCC, i.e. 180° LCP.This is because the SiN ring resonator, in the embodiment depicted, has52 wavelengths in a round trip near 1550 nm free-space wavelength, so⅛^(th) of a wavelength tilt is 360°/52/8˜0.86°.

A physical realization of a 4^(th)-order standard SCC filter 190 isdepicted in FIGS. 19 a-c for comparison. It is a standard Chebyshevfilter, realized in a SCC-type structure. A large physical coupling gapof about 5 microns is introduced between rings 191 and 194 in order tomake the coupling between rings 191 and 194 negligible, so that thestructure 190 may be considered a SCC structure.

The devices 70, 190 depicted in FIGS. 18 a-c and 19 a-c, respectively,were simulated for comparison. Simulated spectra are shown in FIG. 20without excess waveguide loss, and in FIG. 21 with the effect ofwaveguide loss added, corresponding to a loss Q for each cavity of about40,000, which is a realizable loss Q in many material systems. In thefigure legends, spectral responses of the SCC structure 190 are markedas ‘SCC’, while those of the loop-coupled design 70 of the presentinvention are marked as “loop-coupled”. FIG. 20 shows that for the samethrough-port extinction band of −30 dB across 40 nm, the loop-coupleddevice 70 has faster rolloff, as is also shown in FIG. 12. FIG. 21 showsthat the introduction of transmission zeros permits the passband of theloop-coupled filter 70 to widen and lower the drop-port group delayslightly such that, for the same resonator losses, the filter drop lossis lower in the loop-coupled structure 70 than in the SCC structure 190.

FIGS. 22-25 illustrate another advantage that may be attained in opticalsignal processing by using a loop coupled structure. FIG. 23 shows thedrop (dashed line 232) and through (solid line 234) response of a4^(th)-order Chebyshev filter, using a SCC structure, designed to have a40 GHz passband with 30 dB in-band extinction in the through port,and >30 dB rejection in the drop port at adjacent channels, with 100 GHzchannel spacing.

FIG. 22 shows the drop (thick solid line 222) and through (thicklong-dashed line 224) port responses of the loop-coupled structure 70depicted in FIG. 7, where the transmission nulls were not used to make asharper rolloff, but rather to widen, flatten, and round the edges ofthe passband. In such a fashion, the drop-port passband resembles amaximally-flat, Butterworth response, while the stopband of the dropport resembles a sharper rolloff due to zeros. This is similar to theinverse-Chebyshev (sometimes called Chebyshev Type II) filter response.Also shown for comparison in FIG. 22 are the drop (thin short-dashedline 226) and through (thin dash-dot line 228, which overlaps with theother through port 224) responses of a standard Butterworth(maximally-flat) all-pole filter, implementable in a SCC structure. Ithas the same passband shape, but shows clearly an insufficient rolloffrate, in comparison, to meet 30 dB rejection at the adjacent channel.

FIG. 24 compares the group delay spectra (same for the drop ports andthrough ports) of the 4^(th)-order Chebyshev filter (dashed line 242 inFIG. 24), whose amplitude response is illustrated in FIG. 23, to that ofthe loop-coupled structure 70 (solid line 244 in FIG. 24) depicted inFIG. 7. FIG. 25 compares the dispersion spectra (valid for the drop portand the through port responses) of the 4^(th)-order Chebyshev filter(dashed line 252) to that of the loop-coupled structure 70 (solid line254). The loop-coupled structure 70 shows less than 12 ps group delayin-band to the drop port across 40 GHz (FIG. 24), and less than 7 ps/nmin-band dispersion in the drop port (FIG. 25), as well as less than 7ps/nm out-of-band dispersion in the through port at the edge of theadjacent channel, this being 80 GHz away from the channel centerassuming 100 GHz channel spacing and a 40 GHz spectral width for eachchannel (FIG. 25). On the other hand, the standard Chebyshev filtershows 15-25 ps group delay in-band (FIG. 24), with 43 ps/nm maximumdispersion in the drop port in-band (FIG. 25), and 3 ps/nm maximumdispersion at the edge of the adjacent channel in the through port (FIG.25). Therefore, for a slight increase in adjacent channel through portdispersion, the loop-coupled filter 70 reduces the drop-port dispersionby a factor of 6, while meeting the same spectral amplitude criteria, bypermitting the widening of the passband. This is one approach by whichloop-coupled filters may reduce filter dispersion. An alternative way,by making a phase-linearizing filter, was already described withreference to FIGS. 13-15.

The approach for obtaining N transmission zeros (per FSR) in an N-cavitysystem by coupling the input and output waveguides directly at a powerfraction equal to the desired drop-port transmission level at largedetunings is valid generally. More generally, the number of transmissionzeros in the drop port is equal to N−M, if N is the number of cavities,and M is the smallest number of coupled cavities that must be traversedin going from the input port to the drop port.

FIG. 26 depicts one embodiment of another inventive device 260. Thedevice 260 includes a standard SCC second-order resonator 261 couplingan input waveguide 263 to an output waveguide 264, followed by adirectional coupler 262 optically coupling the input waveguide 263 tothe output waveguide 264. Such a design permits the sharpness of thesecond order filter 260 to be increased.

FIG. 27 shows a standard SCC filter drop and through response (lines 271and 274, respectively) and the drop and through responses of the device260 (lines 272 and 273, respectively). By setting the directionalcoupler 262 of the device 260 at about 1% (−20 dB), the drop port 265response rolloff is increased by about 4-5 dB, and the through-port 267extinction is at the same time increased by 5 dB over the SCC filter.The coupling coefficients may be chosen so as to improve more the dropport 265 rolloff rate at the expense of smaller improvements inthrough-port 267 extinction or vice versa. The price paid for theimproved selectivity is that there is no rolloff at large detuning sincethe filter 260 has 2 poles and 2 zeros, and levels off at −20 dB in thedrop port 265. This response design may be accomplished by adding a 90°phase delay 266 in one waveguide 264, 263 in the region between theresonators 261 and the directional coupler 262, relative to the phasedelay in propagating through the other waveguide 263,264 in the regionbetween the resonators 261 and the coupler 262 (the lengths of the inputand output waveguides 263, 264 from the resonators 261 to thedirectional coupler 262 otherwise being of substantially equal length).Filter responses such as the one depicted in FIG. 27 may be employed inchannel add-drop filters in cascade geometries, where at least onefilter is the filter 261 of structure 260, and other filters arestandard SCC filters. By cascading the filter 261 of the structure 260with a standard second-order SCC filter, a drop-port rolloff is obtainedat large frequency detunings, while the rejection improvements achievedby direct coupling prevent substantial power loss in the first filterstage at the dropped or adjacent channel. This can be useful forproviding multistage telecom-grade filter designs using only 2-ringfilter stages, which are simpler to fabricate than higher order devices.

In one embodiment, the optical coupling between the input waveguide 263and the output waveguide 264 introduced by the directional coupler 262is substantially broadband over several passband widths. For example,the optical coupling may be substantially broadband over at least threepassband widths, or over at least ten passband widths. In addition, itshould be noted that any number of resonators 261 may be included in thefilter 260, including a single resonator 261 or three or more resonators261.

Referring to FIG. 28, a loop-coupled resonator based optical filter 280is shown comprising four standing-wave resonators 281-284. An opticalhybrid structure 290, with similar performance to the structure 70 basedon loop-coupled ring resonators (depicted in FIG. 7), can be createdusing standing wave resonators as shown in FIG. 29. It is a combinationof two of the resonant structures 280 shown in FIG. 28, with a mirrorsymmetry plane 297 between them as shown. In addition, to create matchedports, resonators 291 and 291′ are coupled to each other and to an inputwaveguide 295 with separated input and through ports, and resonators 294and 294′ are coupled to each other and to an output waveguide 296 withseparated drop and add ports. If the structure in FIG. 28 has a positiveLCC (zero LCP), then the structure in FIG. 29 may be an optical hybridwith likewise a positive LCC (zero LCP).

In FIG. 30, by coupling resonator 301 to 304′, and 301′ to 304,respectively, instead of 301 to 304 and 301′ to 304′ respectively, onemay change the LCC from positive to negative (i.e. 180° LCP). In FIG.31, by coupling resonator 341 to 344′, and 341′ to 344, respectively, aswell as 341 to 341′, and 344 to 344′, respectively, one may obtainarbitrary LCP values in the optical hybrid device 340. Such arbitrary(i.e., not zero or 180° LCP values) may not be typically obtained in astanding-wave structure using reciprocal (i.e., no use of magnetoopticmedia) resonators in FIG. 28, while it may be obtained in FIG. 31 withreciprocal resonators. Arbitrary LCP was also shown to be achievable inembodiments of the invention that include ring resonator loop-coupledstructures, such as the structure 70 depicted in FIGS. 7 and 8 and moregenerally shown in FIG. 6.

FIG. 32 shows a loop-coupled resonator based optical filter 350comprising four cavities 351-354 each supporting a higher-order mode,and showing positive LCC (zero LCP). FIG. 33 shows a loop-coupledresonator based optical filter 360 with cavity 363 oriented so that thestructure 360 has a negative LCC (180° LCP).

FIG. 34 shows an optical hybrid structure 370 that may show the sametransmission characteristics as the loop-coupled structure 350 depictedin FIG. 32. It includes two of the loop-coupled structures 350 of FIG.32, but has resonators 371 and 371′ coupled to each other and to aninput waveguide 375 with separated ports, and has resonators 374 and374′ coupled to each other and to an output waveguide 376 with separatedports to facilitate optical hybrid operation.

FIG. 35 shows an optical hybrid structure 380 that may show the sametransmission characteristics as the loop-coupled structure 360 depictedin FIG. 33, i.e., the behavior of a loop-coupled structure with a singlecoupling loop having a negative LCC (180° LCP). An alternative way toform a negative LCC, using, for example, the cavity modes ofloop-coupled structure 350 depicted in FIG. 32, is to replace to cavitycouplings 381-384 and 381′-384′ with couplings 381-384′ and 381′-384, asshown for cavities 391, 391′, 394, and 394′ in FIG. 36.

FIG. 37 shows an optical hybrid structure 400 that may show the sametransmission characteristics as the loop-coupled structure 70 depictedin FIG. 7. In addition, the structure 400 may have arbitrary LCP, whichmay not typically be achieved in either structure 350 depicted in FIG.32 or in structure 360 depicted in FIG. 33 if those structures includeexclusively electromagnetically reciprocal resonators, as is typicallythe case.

Optical hybrid structures may similarly be created from otherloop-coupled standing-wave cavity structures by converting each of theinput and output waveguides from a waveguide that ends at the resonator,having substantially high reflection (ideally approaching 100%) whennone of the resonators are excited (and thus having the through port asthe reflected wave relative to the input port), to a waveguide withseparated ports having no substantial reflection, and using two copiesof the standing-wave loop coupled structure, the two copies beingcoupled to each other at those cavities that are coupled to accesswaveguides, and optionally having other cross-coupling. The opticalhybrid, besides having a non-reflecting waveguide, in general has twiceas many coupling loops as the non-hybrid standing-wave structure. Thus,the embodiments provided in the invention describe standing-wavestructures, such as the structure 350 depicted in FIG. 32, having atleast one coupling loop, and standing-wave-resonator optical hybrids(coupled at least twice to each waveguide) accordingly having at leasttwo coupling loops. In one embodiment, in standing-wave-resonatoroptical hybrids, the double coupling to each waveguide is done toappropriately to set up traveling-wave-like excitation from thewaveguide, i.e., the excitation of two modes with degenerate frequencyand 90° out of phase, to simulate ring resonator operation. Thesedesigns may be used to produce an operation similar to a microringfilter, such as that shown in FIG. 7, using, for example, photoniccrystal (PhC) standing-wave microcavities. In various embodiments, theloop-coupled structures provided may be configured to have at least onecoupling loop with zero, 180°, or another value of the loop-couplingphase (LCP), thereby permitting the engineering of transmission zeros asalready illustrated in the given examples.

In summary, the structures presented herein enable filter responses thathave, in certain embodiments, transmission-response (drop-port) zeros atcomplex-frequency detunings. The structures may include an inputwaveguide, at least one output waveguide, and a coupled-cavity systemcoupling the input and output waveguides. The coupled-cavity system mayinclude cavity coupling loops, each with a loop-coupling phasedetermined by the geometry of the coupling loop. The loop coupling phasemay be approximately 0 or 180° for standing wave resonators usinghigh-order spatial modes, and may take on arbitrary values forreciprocal microring resonator structures operated with unidirectionalexcitation, or when standing- or traveling-wave resonators made ofnon-reciprocal (magnetooptic) media are used. By designing couplingloops with LCPs, and with the optional addition of a direct,phase-aligned coupling between the input and output port, the control ofN poles and N zeros (per FSR) may be obtained in an N cavity filter.This allows for the design of optimum filters—in the sense of optimallysharp amplitude responses and non-minimum-phase designs permittinglinear-phase or phase-engineered passbands—and for optical signalprocessing structures in a compact and robust implementation.

Having described certain embodiments of the invention, it will beapparent to those of ordinary skill in the art that other embodimentsincorporating the concepts disclosed herein may be used withoutdeparting from the spirit and scope of the invention. Accordingly, thedescribed embodiments are to be considered in all respects as onlyillustrative and not restrictive.

1-18. (canceled)
 19. An optical resonator structure, comprising: aninput waveguide; an output waveguide; at least one resonator couplingthe input waveguide to the output waveguide; and a directional coupleroptically coupling the input waveguide to the output waveguide.
 20. Thestructure of claim 19 further comprising a phase shift in the inputwaveguide between the directional coupler and the at least oneresonator.
 21. The structure of claim 19 further comprising a phaseshift in the output waveguide between the directional coupler and the atleast one resonator.
 22. The structure of claim 19 wherein a length ofthe input waveguide between the directional coupler and a point at whichthe at least one resonator couples to the input waveguide issubstantially equal to a length of the output waveguide between thedirectional coupler and a point at which the at least one resonatorcouples to the output waveguide.
 23. The structure of claim 19, whereina plurality of resonators couple the input waveguide to the outputwaveguide.
 24. The structure of claim 23, wherein the plurality ofresonators comprise a sequence of resonators that form a coupling loop,each resonator in the sequence coupled to at least two other resonatorsin the sequence and a first resonator in the sequence coupled to a lastresonator in the sequence so as to form the coupling loop.
 25. Thestructure of claim 19, wherein the optical coupling between the inputwaveguide and the output waveguide introduced by the directional coupleris substantially broadband over several passband widths.
 26. Thestructure of claim 25, wherein the optical coupling is substantiallybroadband over at least three passband widths.
 27. The structure ofclaim 25, wherein the optical coupling is substantially broadband overat least ten passband widths.
 28. The structure of claim 19, wherein theoutput waveguide comprises a drop port and the optical resonatorstructure comprises a spectral response with transmission zeros in thedrop port.